# Uniform integrability and Lebesgue convergence

A). Given that $|X_n| \leq Y$ and $Y \in L$. Try to show $X_n$ is lebesgue integrable.

b). Try to give any example for which $X_n \longrightarrow^{L} X$ yet $\not\exists Y \in L$ with $|X_n| \leq Y$.

c). Try to give any example for which $X_n \to X$ w.p.1, and $X_n, X \in L$ yet $X_n \not\longrightarrow^L X$.

d). If $X_n$ is uniform integrable, does it follow that $g(X_n)$ is uniform integrable if g is continuous?

-
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – Did Oct 28 '12 at 8:23
Maybe $L$ denote the set of integrable random variables defined on a probability space. In b), you have to give more precisions about what $X_n\to X$ means (in which sense), even if we can guess it. – Davide Giraudo Oct 28 '12 at 10:00
Dear David, I have rewrite my description. In part b) what I mean is $X_n \longrightarrow^L X$. – abrocod Oct 28 '12 at 17:48

1. There is no need of the parameter $n$: if $0\leq X\leq Y$ and $Y$ is integrable, it follows from the definition of Lebesgue integral that $X$ is integrable. If $0\leq s\leq X$ is a simple function, then $0\leq s\leq Y$ so $\sup\{\int S,0\leq S\leq X,S\mbox{ simple}\}$ is finite.

2. Try $X_n:=\sqrt n\chi_{((n+1)^{—1},n^{-1})}$.

3. Try $X_n:=n\chi_{((n+1)^{—1},n^{-1})}$.

4. Take $f$ an integrable function which is not in $L^2$. Then $X_n(x):=f(x+n)$ and $g(x)=x^2$.

-
Thanks for your answer, David. I have two wonderings: Is $\chi$ in part 2 means chi-square density? And how to show g(x) is not uniform integrable in part 4? – abrocod Oct 28 '12 at 22:50
No, it's the characteristic function of a set. You have to show that $\{f(\cdot +n)^2\}$ is not uniformly integrable. If it was, it would be bounded $L^1$, so each function would be in $L^1$. – Davide Giraudo Oct 28 '12 at 22:52
hi, david, can you please be more specific about what does $\chi_{(n+1)^{-1}, n^{-1}}$ mean? – abrocod Oct 29 '12 at 1:20
Sorry, parenthesis where missing. $\chi_A(x)$ is $1$ if $x\in A$ and $0$ otherwise. – Davide Giraudo Oct 29 '12 at 9:15